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AP Calculus AB : Slope of a curve at a point

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Example Questions

Example Question #121 : Derivatives

$\begin{align*}&\text{Calculate the slope of the function:}\\&f(x)=4x + cos(8\cdot \pi x) - sin(\frac{(13\cdot \pi x)}{2}) + 2x^{2} + 3\\&\text{At the point }x=2\end{align*}$

Possible Answers:

-627.65

32.42

Correct answer:

32.42

Explanation:

$\begin{align*}&\text{Recall the definition of slope as rise over run. To put it in}\\&\text{other words, it is a change in height in relation to a change}\\&\text{horizontal position: it is a rate. And since it is a rate, it}\\&\text{can be found with a derivative. In this case, the derivative}\\&\text{of our function (at a height at a given point) with respect}\\&\text{to x:}\\&Slope=\frac{df(x)}{dx}\\&\text{Taking the derivative of our function, }4x + cos(8\cdot \pi x) - sin(\frac{(13\cdot \pi x)}{2}) + 2x^{2} + 3\\&\text{We find the slope function: }4x -\frac{ (13\cdot \pi cos(\frac{(13\cdot \pi x)}{2}))}{2}- 8\cdot \pi sin(8\cdot \pi x) + 4\\&\text{Plugging in our value of x we then get:}\\&f'(2)=4(2) -\frac{ (13\cdot \pi cos(\frac{(13\cdot \pi (2))}{2}))}{2}- 8\cdot \pi sin(8\cdot \pi (2)) + 4\\&f'(2)=32.42\end{align*}$

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Example Question #22 : Derivative At A Point

$\begin{align*}&\text{Calculate the slope of the function:}\\&f(x)=cos(\frac{(13\cdot \pi x)}{2}) + 2sin(4\cdot \pi x) + 2\\&\text{At the point }x=5\end{align*}$

Possible Answers:

0.4

-0.2

4.71

Correct answer:

4.71

Explanation:

$\begin{align*}&\text{Recall the definition of slope as rise over run. To put it in}\\&\text{other words, it is a change in height in relation to a change}\\&\text{horizontal position: it is a rate. And since it is a rate, it}\\&\text{can be found with a derivative. In this case, the derivative}\\&\text{of our function (at a height at a given point) with respect}\\&\text{to x:}\\&Slope=\frac{df(x)}{dx}\\&\text{Taking the derivative of our function, }cos(\frac{(13\cdot \pi x)}{2}) + 2sin(4\cdot \pi x) + 2\\&\text{We find the slope function: }8\cdot \pi cos(4\cdot \pi x) -\frac{ (13\cdot \pi sin(\frac{(13\cdot \pi x)}{2}))}{2}\\&\text{Plugging in our value of x we then get:}\\&f'(5)=8\cdot \pi cos(4\cdot \pi (5)) -\frac{ (13\cdot \pi sin(\frac{(13\cdot \pi (5))}{2}))}{2}\\&f'(5)=4.71\end{align*}$

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Example Question #23 : Derivative At A Point

$\begin{align*}&\text{For positive values of x: }\\&f(x)=2sin(4\cdot \pi x) - cos(4\cdot \pi x) - 5x - 2x^{2} - 3x^{3} + 3\\&\text{Calculate the slope of the function at }x=2\end{align*}$

Possible Answers:

117.91

-23.87

Correct answer:

-23.87

Explanation:

$\begin{align*}&\text{Recall the definition of slope as rise over run. To put it in}\\&\text{other words, it is a change in height in relation to a change}\\&\text{horizontal position: it is a rate. And since it is a rate, it}\\&\text{can be found with a derivative. In this case, the derivative}\\&\text{of our function (at a height at a given point) with respect}\\&\text{to x:}\\&Slope=\frac{df(x)}{dx}\\&\text{Taking the derivative of our function, }2sin(4\cdot \pi x) - cos(4\cdot \pi x) - 5x - 2x^{2} - 3x^{3} + 3\\&\text{We find the slope function: }8\cdot \pi cos(4\cdot \pi x) - 4x + 4\cdot \pi sin(4\cdot \pi x) - 9x^{2} - 5\\&\text{Plugging in our value of x we then get:}\\&f'(2)=8\cdot \pi cos(4\cdot \pi (2)) - 4(2) + 4\cdot \pi sin(4\cdot \pi (2)) - 9(2)^{2} - 5\\&f'(2)=-23.87\end{align*}$

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Example Question #24 : Derivative At A Point

$\begin{align*}&\text{Find the slope of: }\\&f(x)=x + cos(\frac{(3\cdot \pi x)}{2}) + 4cos(\frac{(3\cdot \pi x)}{4}) - sin(7\cdot \pi x) - x^{2} + 3x^{3} - 2\\&\text{At }x=2\end{align*}$

Possible Answers:

56.21

20.43

9.5

Correct answer:

20.43

Explanation:

$\begin{align*}&\text{Recall the definition of slope as rise over run. To put it in}\\&\text{other words, it is a change in height in relation to a change}\\&\text{horizontal position: it is a rate. And since it is a rate, it}\\&\text{can be found with a derivative. In this case, the derivative}\\&\text{of our function (at a height at a given point) with respect}\\&\text{to x:}\\&Slope=\frac{df(x)}{dx}\\&\text{Taking the derivative of our function, }x + cos(\frac{(3\cdot \pi x)}{2}) + 4cos(\frac{(3\cdot \pi x)}{4}) - sin(7\cdot \pi x) - x^{2} + 3x^{3} - 2\\&\text{We find the slope function: }9x^{2} - 7\cdot \pi cos(7\cdot \pi x) -\frac{ (3\cdot \pi sin(\frac{(3\cdot \pi x)}{2}))}{2}- 3\cdot \pi sin(\frac{(3\cdot \pi x)}{4}) - 2x + 1\\&\text{Plugging in our value of x we then get:}\\&f'(2)=9(2)^{2} - 7\cdot \pi cos(7\cdot \pi (2)) -\frac{ (3\cdot \pi sin(\frac{(3\cdot \pi (2))}{2}))}{2}- 3\cdot \pi sin(\frac{(3\cdot \pi (2))}{4}) - 2(2) + 1\\&f'(2)=20.43\end{align*}$

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Example Question #31 : Derivative At A Point

$\begin{align*}&\text{Find the slope of: }\\&f(x)=4x + cos(9\cdot \pi x) - cos(\frac{(11\cdot \pi x)}{2}) + 5x^{2} + x^{3} - 4\\&\text{At }x=2\end{align*}$

Possible Answers:

-1075.99

Correct answer:

Explanation:

$\begin{align*}&\text{Recall the definition of slope as rise over run. To put it in}\\&\text{other words, it is a change in height in relation to a change}\\&\text{horizontal position: it is a rate. And since it is a rate, it}\\&\text{can be found with a derivative. In this case, the derivative}\\&\text{of our function (at a height at a given point) with respect}\\&\text{to x:}\\&Slope=\frac{df(x)}{dx}\\&\text{Taking the derivative of our function, }4x + cos(9\cdot \pi x) - cos(\frac{(11\cdot \pi x)}{2}) + 5x^{2} + x^{3} - 4\\&\text{We find the slope function: }10x - 9\cdot \pi sin(9\cdot \pi x) +\frac{ (11\cdot \pi sin(\frac{(11\cdot \pi x)}{2}))}{2}+ 3x^{2} + 4\\&\text{Plugging in our value of x we then get:}\\&f'(2)=10(2) - 9\cdot \pi sin(9\cdot \pi (2)) +\frac{ (11\cdot \pi sin(\frac{(11\cdot \pi (2))}{2}))}{2}+ 3(2)^{2} + 4\\&f'(2)=36\end{align*}$

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Example Question #32 : Derivative At A Point

$\begin{align*}&\text{Calculate the slope of the function:}\\&f(x)=4x - cos(\frac{(17\cdot \pi x)}{2}) + 3sin(2\cdot \pi x) - sin(6\cdot \pi x) - x^{2} - 3x^{3} - 4\\&\text{At the point }x=9\end{align*}$

Possible Answers:

-248.44

-716.3

-247.89

Correct answer:

-716.3

Explanation:

$\begin{align*}&\text{Recall the definition of slope as rise over run. To put it in}\\&\text{other words, it is a change in height in relation to a change}\\&\text{horizontal position: it is a rate. And since it is a rate, it}\\&\text{can be found with a derivative. In this case, the derivative}\\&\text{of our function (at a height at a given point) with respect}\\&\text{to x:}\\&Slope=\frac{df(x)}{dx}\\&\text{Taking the derivative of our function, }4x - cos(\frac{(17\cdot \pi x)}{2}) + 3sin(2\cdot \pi x) - sin(6\cdot \pi x) - x^{2} - 3x^{3} - 4\\&\text{We find the slope function: }6\cdot \pi cos(2\cdot \pi x) - 2x - 6\cdot \pi cos(6\cdot \pi x) +\frac{ (17\cdot \pi sin(\frac{(17\cdot \pi x)}{2}))}{2}- 9x^{2} + 4\\&\text{Plugging in our value of x we then get:}\\&f'(9)=6\cdot \pi cos(2\cdot \pi (9)) - 2(9) - 6\cdot \pi cos(6\cdot \pi (9)) +\frac{ (17\cdot \pi sin(\frac{(17\cdot \pi (9))}{2}))}{2}- 9(9)^{2} + 4\\&f'(9)=-716.3\end{align*}$

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Example Question #31 : Derivative At A Point

$\begin{align*}&\text{Calculate the slope of the function:}\\&f(x)=ln(3x) + sin(\frac{(9\cdot \pi x)}{2}) + 2\\&\text{At the point }x=6\end{align*}$

Possible Answers:

-13.97

0.82

-0.03

$\infty$

Correct answer:

-13.97

Explanation:

$\begin{align*}&\text{Recall the definition of slope as rise over run. To put it in}\\&\text{other words, it is a change in height in relation to a change}\\&\text{horizontal position: it is a rate. And since it is a rate, it}\\&\text{can be found with a derivative. In this case, the derivative}\\&\text{of our function (at a height at a given point) with respect}\\&\text{to x:}\\&Slope=\frac{df(x)}{dx}\\&\text{Taking the derivative of our function, }ln(3x) + sin(\frac{(9\cdot \pi x)}{2}) + 2\\&\text{We find the slope function: }\frac{(9\cdot \pi cos(\frac{(9\cdot \pi x)}{2}))}{2}+\frac{ 1}{x}\\&\text{Plugging in our value of x we then get:}\\&f'(6)=\frac{(9\cdot \pi cos(\frac{(9\cdot \pi (6))}{2}))}{2}+\frac{ 1}{(6)}\\&f'(6)=-13.97\end{align*}$

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Example Question #33 : Derivative At A Point

$\begin{align*}&\text{Calculate the slope of the function:}\\&f(x)=2ln(3x) - 4x + sin(5\cdot \pi x) - sin(\frac{(9\cdot \pi x)}{2}) + 3x^{2} + 6x^{3} + 2\\&\text{At the point }x=1\end{align*}$

Possible Answers:

8.2

6.29

239.86

$\infty$

Correct answer:

6.29

Explanation:

$\begin{align*}&\text{Recall the definition of slope as rise over run. To put it in}\\&\text{other words, it is a change in height in relation to a change}\\&\text{horizontal position: it is a rate. And since it is a rate, it}\\&\text{can be found with a derivative. In this case, the derivative}\\&\text{of our function (at a height at a given point) with respect}\\&\text{to x:}\\&Slope=\frac{df(x)}{dx}\\&\text{Taking the derivative of our function, }2ln(3x) - 4x + sin(5\cdot \pi x) - sin(\frac{(9\cdot \pi x)}{2}) + 3x^{2} + 6x^{3} + 2\\&\text{We find the slope function: }6x + 5\cdot \pi cos(5\cdot \pi x) -\frac{ (9\cdot \pi cos(\frac{(9\cdot \pi x)}{2}))}{2}+\frac{ 2}{x}+ 18x^{2} - 4\\&\text{Plugging in our value of x we then get:}\\&f'(1)=6(1) + 5\cdot \pi cos(5\cdot \pi (1)) -\frac{ (9\cdot \pi cos(\frac{(9\cdot \pi (1))}{2}))}{2}+\frac{ 2}{(1)}+ 18(1)^{2} - 4\\&f'(1)=6.29\end{align*}$

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Example Question #33 : Derivative At A Point

$\begin{align*}&\text{For positive values of x: }\\&f(x)=3x^{3} - cos(7\cdot \pi x) - 6\\&\text{Calculate the slope of the function at }x=2\end{align*}$

Possible Answers:

8.5

519.61

Correct answer:

Explanation:

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Example Question #131 : Derivatives

$\begin{align*}&\text{Calculate the slope of the function:}\\&f(x)=3sin(2\cdot \pi x) - 3x - sin(\frac{(\pi x)}{2}) + 2\\&\text{At the point }x=9\end{align*}$

Possible Answers:

-2.89

15.85

2.47

-3.11

Correct answer:

15.85

Explanation:

$\begin{align*}&\text{Recall the definition of slope as rise over run. To put it in}\\&\text{other words, it is a change in height in relation to a change}\\&\text{horizontal position: it is a rate. And since it is a rate, it}\\&\text{can be found with a derivative. In this case, the derivative}\\&\text{of our function (at a height at a given point) with respect}\\&\text{to x:}\\&Slope=\frac{df(x)}{dx}\\&\text{Taking the derivative of our function, }3sin(2\cdot \pi x) - 3x - sin(\frac{(\pi x)}{2}) + 2\\&\text{We find the slope function: }6\cdot \pi cos(2\cdot \pi x) -\frac{ (\pi cos(\frac{(\pi x)}{2}))}{2}- 3\\&\text{Plugging in our value of x we then get:}\\&f'(9)=6\cdot \pi cos(2\cdot \pi (9)) -\frac{ (\pi cos(\frac{(\pi (9))}{2}))}{2}- 3\\&f'(9)=15.85\end{align*}$

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AP Calculus AB : Slope of a curve at a point

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #121 : Derivatives

Example Question #22 : Derivative At A Point

Example Question #23 : Derivative At A Point

Example Question #24 : Derivative At A Point

Example Question #31 : Derivative At A Point

Example Question #32 : Derivative At A Point

Example Question #31 : Derivative At A Point

Example Question #33 : Derivative At A Point

Example Question #33 : Derivative At A Point

Example Question #131 : Derivatives

All AP Calculus AB Resources

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